linear fractional transformation
[P1]=lft(P,K)
[P1]=lft(P,r,K)
[P1,r1]=lft(P,r,Ps,rs)
: :r 1x2 row vector, dimension of P22 : :Ps linear system ( syslin list), implicitly partitioned into four
blocks (two input ports and two output ports).
: :rs 1x2 row vector, dimension of Ps22 :
Linear fractional transform between two standard plants P and Ps in state space form or in transfer form ( syslin lists).
r= size(P22) rs=size(P22s)
lft(P,r, K) is the linear fractional transform between P and a controller K ( K may be a gain or a controller in state space form or in transfer form);
lft(P,K) is lft(P,r,K) with r`=size of `K transpose;
P1= P11+P12*K* (I-P22*K)^-1 *P21
[P1,r1]=lft(P,r,Ps,rs) returns the generalized (2 ports) lft of P and Ps.
P1 is the pair two-port interconnected plant and the partition of P1 into 4 blocks in given by r1 which is the dimension of the 22 block of P1.
P and R can be PSSDs i.e. may admit a polynomial D matrix.
s=`poly`_(0,'s');
P=[1/s, 1/(s+1); 1/(s+2),2/s]; K= 1/(s-1);
lft(P,K)
lft(P,[1,1],K)
P(1,1)+P(1,2)*K*`inv`_(1-P(2,2)*K)*P(2,1) //Numerically dangerous!
`ss2tf`_(lft(`tf2ss`_(P),`tf2ss`_(K)))
lft(P,-1)
f=[0,0;0,1];w=P/.f; w(1,1)
//Improper plant (PID control)
W=[1,1;1,1/(s^2+0.1*s)];K=1+1/s+s
lft(W,[1,1],K); `ss2tf`_(lft(`tf2ss`_(W),[1,1],`tf2ss`_(K)))